Conjecture on short admissible profiles in the hypercube

Let $Q^n$ be the $n$-dimensional hypercube, and let an admissible tuple be a tuple that occurs as the profile of a matching in $Q^n$. For $\bm{x}=(x_1,\ldots,x_n)\in\mathbb{N}^n$, write $||\bm{x}||_1=\sum_{i=1}^n x_i$. Short-profile admissibility conjecture. Every tuple $\bm{x}\in\mathbb{N}^n$ satisfying

$$||\bm{x}||_1<2^{n-1}$$

is admissible. The claim extends the verified computations for $n\leq 4$ and proposes that, below the size of a perfect matching, there are no restrictions beyond the definition of a profile.